Problem 50 - Calculus Single Variable

Question

In the table, \(x\) represents cigarette consumption per adult (in hundreds) for 1962 for eight countries. The variable \(y\) represents mortality per 100,000 due to heart disease in 1962 $$ \begin{array}{|l|c|c|} \hline & \boldsymbol{x} & \boldsymbol{y} \\ \hline \text { Australia } & 32.2 & 238.1 \\ \hline \text { Belgium } & 17.0 & 118.1 \\ \hline \text { Canada } & 33.5 & 211.6 \\ \hline \text { West Germany } & 18.9 & 150.3 \\ \hline \text { Ireland } & 27.7 & 187.3 \\ \hline \text { Netherlands } & 18.1 & 124.7 \\ \hline \text { Great Britain } & 27.9 & 194.1 \\ \hline \text { United States } & 39 & 256.9 \\ \hline \end{array} $$ Plot the eight points \((x, y)\). Find a function \(f(x)=\) \(m x+12.5\) that could be used to model the relationship between cigarette consumption and mortality due to heart disease.

Step-by-Step Solution

Answer

Plot the points and estimate the slope \(m\) as \(8.24\). The linear model is \(f(x) = 8.24x + 12.5\).

Step 1: Prepare the Coordinates

Extract the given values of \(x\) and \(y\) for each country from the table. These are: \((32.2, 238.1)\), \((17.0, 118.1)\), \((33.5, 211.6)\), \((18.9, 150.3)\), \((27.7, 187.3)\), \((18.1, 124.7)\), \((27.9, 194.1)\), and \((39, 256.9)\).

Step 2: Plot the Points

On graph paper or a plotting software, plot each point with \(x\) on the horizontal axis (representing cigarette consumption per adult) and \(y\) on the vertical axis (representing mortality due to heart disease).

Step 3: Model the Data Using a Linear Function

Given the function \(f(x)=mx+12.5\), we need to determine an appropriate slope \(m\) by considering the general trend of the plotted points. This can be done either by calculating using the least squares method or estimating visually.

Step 4: Apply a Method to Find the Slope

To estimate the slope \(m\), calculate it using two points that seem to fit well as a linear model. A simplified approach can use the averages of \(x\) and \(y\) or select two data points. For example, using Canada \((33.5, 211.6)\) and United States \((39, 256.9)\) yields: \(m = \frac{256.9-211.6}{39-33.5} = \frac{45.3}{5.5} \approx 8.24\).

Step 5: Write the Linear Model

Substitute the slope \(m\) back into the function \(f(x) = mx + 12.5\). With \(m = 8.24\), the function becomes \(f(x) = 8.24x + 12.5\).

Key Concepts

Scatter Plot

In the world of data analysis, a scatter plot is a powerful tool. It's a type of graph that uses dots to represent the values obtained for two different variables. By plotting the points from our dataset, we can easily visualize the relationship between cigarette consumption and heart disease mortality.
To create a scatter plot, you'll place the cigarette consumption values on the x-axis and the mortality rates on the y-axis.
Each country provides a pair of values \(x, y\). When you look at the graph, you can observe how changes in one variable correspond with changes in another.

• If the points seem to rise together, it hints at a positive correlation.
• If one increases while the other decreases, it suggests a negative correlation.
• Clusters of points may indicate specific patterns or trends.

Scatter plots lay the foundation for linear regression by visually showcasing the relationship between variables.

Slope Calculation

To understand how two variables are related, calculating the slope in a linear regression model is essential. The slope, often denoted as \(m\), represents how much the dependent variable (in our case, mortality rate) changes for a unit change in the independent variable (cigarette consumption).
When determining the slope, we aim to find the best linear approximation of the given data.

• The formula used is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates from the dataset.
• Choosing different points may result in slightly varied slopes, so it's advisable to use multiple methods for accuracy or make use of statistical software for precision.
• In this example, using Canada and United States values resulted in a slope of \(m \approx 8.24\).

Understanding the slope gives insight into the change rate in mortality with changes in smoking habits.

Cigarette Consumption Data

Let's delve deeper into the world of cigarette consumption data. For this analysis, cigarette consumption is quantified as the number of cigarettes smoked per adult, expressed in hundreds. This simplification helps manage and compare large numbers across different countries.
The data for 1962 spans eight countries, showcasing varying levels of consumption.

• Australia leads with 32.2 hundreds of cigarettes per adult.
• Conversely, Belgium shows a lower consumption with only 17.0 hundreds.
• Each country's consumption forms one part of the dataset, providing insight into global smoking trends of the time.

This form of data is crucial when examining its impacts on societal health metrics, such as heart disease mortality.

Heart Disease Mortality Analysis

Understanding the impact of cigarette consumption requires a close look at heart disease mortality rates. Heart disease was a significant health concern globally in 1962, and this analysis explores how lifestyle choices like smoking contribute to this condition.
Mortality rate is given per 100,000 individuals, making it easier to compare across countries:

• The United States had the highest rate at 256.9 deaths per 100,000.
• The lowest was Belgium, with 118.1 deaths per 100,000.
• These rates indicate potential health outcomes associated with different levels of smoking.

The insights gained from such mortality analysis are vital for public health planning and policy-making. As we explore further, this context helps us understand the implications of the linear function derived for cigarette consumption and mortality.